The abelian sandpile models were firstly introduced in 1987 by three physicists, Bak, Tang,
and Wiesenfeld [1]}, who studied it mainly on the integer grid graphs. In 1990, Dhar[2]} generalized their model from grids to arbitrary graphs.
The abelian sandpile model of Dhar begins with a
connected graph \(G = (V, E)\)  and a distinguished vertex \(q\in V \) , called the sink. A configuration of \((G, q)\)  is a vector \(\vec{c}\in \mathbb {N}^{V-q}\) . A non-sink vertex \(v\)  is stable if its degree satisfies \(d(v) > \vec{c}(v)\) ; otherwise it is unstable. Moreover, a configuration is stable
if every vertex \(v\)  in \(V-q\)  is stable. Toppling an unstable vertex \(u\in V-q\)  in \(\vec{c}\)  is the operation performed by decreasing
\(\vec{c}(u)\)  by the degree \(d(u)\) , and for each neighbour \(v\)  of \(u\)  different from \(q\) , adding the multiplicity \(m(u,v)\)  of the edge \(uv\)  to \(\vec{c}(v)\) . Starting from any initial configuration \(\vec{c}\) , by performing a sequence of topplings, we eventually arrive at a stable configuration. It is not hard to see that the stabilization of an unstable configuration is
unique [2]}, [4]}. The stable configuration associated to \(\vec{c}\)  will be denoted by \(s(\vec{c})\) .
Now, let \((\vec{c} + \vec{d})(u):= \vec{c}(u) + \vec{d}(u)\)  for all \(u\in V-q\)  and \(\vec{c} \oplus \vec{d }:= s(\vec{c} + \vec{d})\) . A configuration \(\vec{c}\)  is recurrent
if it is stable and there exists a non-zero configuration \(\vec{r}\)  such that \(s(\vec{c}+\vec{r}) = \vec{c}\) . Dhar[2]} proved that the number of recurrent configurations is equal to the number of spanning trees of \(G\) , and that the set of recurrent configurations with \(\oplus \)  as a binary operation forms a finite abelian group, which is called the sandpile group of \(G\) .
Soon after that, it was found that the sandpile group is isomorphic
to a number of `classical' abelian groups associated with graphs, such as the group of components in Arithmetic Geometry [6]}, Jacobian group and Picard group in Algebraic Geometry[7]}, the
determinant group in lattice theory[8]},
the critical group of a dollar game [9]}, [4]}.
